A basic model to understand the tumbling-to-tank-treading (TB-TT) transition has been developed by Keller & Skalak [1]} (KS-model); it considers the motion of a pure viscous ellipsoid in simple shear flow, assuming that the particle does not change shape during its motion.
This model addresses the dependency of the TB-TT transition on the viscosity ratio
\(\lambda \) , while it is not capable of predicting two behaviours experimentally observed by Abkarian et al. [2]}: the dependency of the TB-TT transition on the shear rate \(\dot{\gamma }\) , and the swinging motion. To explain the dependency on \(\dot{\gamma }\) , Abkarian et al. [2]} and Skotheim & Secomb [4]} introduced an elastic energy contribution in the membrane model: in the presence of shear rates \(\dot{\gamma }\) , the flow energy is in part dissipated by viscous friction inside the cell and on the membrane (as in KS-model) and in part stored as elastic energy.
When \(\dot{\gamma }\)  is below a critical value \(\dot{\gamma }_c\) , the injected energy is insufficient to trigger tank-treading motion [4]}, and the cell behaves like a rigid body (i.e., it tumbles). For \(\dot{\gamma }\ge \dot{\gamma }_c\) , the membrane starts to tank-tread and the inclination angle oscillates.
